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Important proofs to remember.

I'm doing A level AQA maths exams next June.

In my spec it does say the important proofs e.g. Proving surds are irrational and primes are infinite.

However today I came across some hard proofs such as proving the sine and cosine rules.
Particularly for people who've sat the exams and have experience with past papers and such are there proofs that ask for a lot?

In Wikipedia there are about 5 different ways to prove the cosine rule. Also there's the uncertianty of what other nasty proofs can they throw at me?

I am anxious as I want to get an A* or A in maths. What's the best advice you can give please?

Thanks in advance,

Mad man
Reply 1
Bump
Reply 2
I didn't do AQA, but I would assume you would need to know the proofs for the sum of arithmetic and geometric series. And also being able to show that the derivative of sin is cos, cos to -sin from first principles
Original post by Mad Man
I'm doing A level AQA maths exams next June.

In my spec it does say the important proofs e.g. Proving surds are irrational and primes are infinite.

However today I came across some hard proofs such as proving the sine and cosine rules.
Particularly for people who've sat the exams and have experience with past papers and such are there proofs that ask for a lot?

In Wikipedia there are about 5 different ways to prove the cosine rule. Also there's the uncertianty of what other nasty proofs can they throw at me?

I am anxious as I want to get an A* or A in maths. What's the best advice you can give please?

Thanks in advance,

Mad man


I suppose it depends on what part of the syllabus youre talking about. A1 in
https://filestore.aqa.org.uk/resources/mathematics/specifications/AQA-7357-SP-2017.PDF
is really about deduction/exhaustion/counter example. There should be no "hard" geometry ... proofs in there. There are plenty of examples of such questions in the usual question banks for this section.

However, E1 (trigonometry) mentions that you should understand and use the sin/cos rules and part of that understanding will be to appreciate how they're derived. A simple proof (for both) is to drop an altitude and use the (interior - acute triangle) two right trianges that have been created to derive the rules. Neither is more that a few lines and both are based on using basic trig/pythagoras in the right triangles. Similarly for the series D4/5 that Skiwi mentions, you can prove the GP formula in a couple of lines by using the fact that a finite series would be recursively defined by
S = a + ar + ar^2 + ... + ar^(n-1) = a + r(a + ar + ... + ar^(n-2)) = a + rS - ar^n = ...
These proofs should provide understanding, its less about remembering them for the exams (A1 proof section).
(edited 1 year ago)
Reply 4
Original post by mqb2766
I suppose it depends on what part of the syllabus youre talking about. A1 in
https://filestore.aqa.org.uk/resources/mathematics/specifications/AQA-7357-SP-2017.PDF
is really about deduction/exhaustion/counter example. There should be no "hard" geometry ... proofs in there. There are plenty of examples of such questions in the usual question banks for this section.

However, E1 (trigonometry) mentions that you should understand and use the sin/cos rules and part of that understanding will be to appreciate how they're derived. A simple proof (for both) is to drop an altitude and use the (interior - acute triangle) two right trianges that have been created to derive the rules. Neither is more that a few lines and both are based on using basic trig/pythagoras in the right triangles. Similarly for the series D4/5 that Skiwi mentions, you can prove the GP formula in a couple of lines by using the fact that a finite series would be recursively defined by
S = a + ar + ar^2 + ... + ar^(n-1) = a + r(a + ar + ... + ar^(n-2)) = a + rS - ar^n = ...
These proofs should provide understanding, its less about remembering them for the exams (A1 proof section).

But what about circle theorems (see pic)
I'm worried that they might throw a curve ball at me with GCSE knowledge which I can vaguely remember.
Original post by Mad Man
But what about circle theorems (see pic)
I'm worried that they might throw a curve ball at me with GCSE knowledge which I can vaguely remember.


That comes under the geometry sin/cos rules and apart from the relation BOC = 2BAC, its very similar to the basic sin rule proof where you drop an altitude. So its not that much different. Note that even if you didnt remember that rule, you could have drawn the radius OA and essentially derived the relationship using isosceles triangles, which is pretty much what most of the circle theorems are based on. So for that textbook(?) question, learn the basic sin/cos rule proofs and run over the ~8 gcse circle theorems now and again.
Reply 6
Original post by mqb2766
That comes under the geometry sin/cos rules and apart from the relation BOC = 2BAC, its very similar to the basic sin rule proof where you drop an altitude. So its not that much different. Note that even if you didnt remember that rule, you could have drawn the radius OA and essentially derived the relationship using isosceles triangles, which is pretty much what most of the circle theorems are based on. So for that textbook(?) question, learn the basic sin/cos rule proofs and run over the ~8 gcse circle theorems now and again.


Sorry what do you mean by drop the altitude?
Original post by Mad Man
Sorry what do you mean by drop the altitude?


In
https://www.mathopenref.com/lawofsinesproof.html
the line h is the altitude. Its OP in your posted question. So drop a line from a vertex so that its perpendicular to the opposing side. Then you can prove the sin rule by using basic trig on the two right triangles to calculate h and equate.
Reply 8
Original post by mqb2766
In
https://www.mathopenref.com/lawofsinesproof.html
the line h is the altitude. Its OP in your posted question. So drop a line from a vertex so that its perpendicular to the opposing side. Then you can prove the sin rule by using basic trig on the two right triangles to calculate h and equate.

What's perpendicular to the opposing side?
Original post by Mad Man
What's perpendicular to the opposing side?


The altitude h makes a right angle (perpendicular) with side a which is opposite vertex A.
Reply 10
Is my working out correct?
16653404032683676861131040164106.jpg
Reply 11
Original post by mqb2766
The altitude h makes a right angle (perpendicular) with side a which is opposite vertex A.


What do you think?
Reply 12
But what makes a/sin(B) not viable? Are the letters arbitrary?
Original post by Mad Man
But what makes a/sin(B) not viable? Are the letters arbitrary?

The sin rule involves the ratio (fraction) of the sin of one angle "Z" and the length of the opposing side "z". You can;t mix and match the sides and angles. You should be familiar with the A,B,C angle names and the corresponding opposite sides a,b,c?
Original post by Mad Man
What do you think?

This must be for the textbook question as the previous link covers the proof of the general sin rule. The way youve drawn the lines passing through the centre of the circumcircles, they'd neither bisect the opposite sides nor meet at right angles. The diagram given in the question is the way to approach it and using the simple circle theorem to relate the two angles together as described in the model solution.
(edited 1 year ago)
Reply 14
Original post by mqb2766
The sin rule involves the ratio (fraction) of the sin of one angle "Z" and the length of the opposing side "z". You can;t mix and match the sides and angles. You should be familiar with the A,B,C angle names and the corresponding opposite sides a,b,c?

This must be for the textbook question as the previous link covers the proof of the general sin rule. The way youve drawn the lines passing through the centre of the circumcircles, they'd neither bisect the opposite sides nor meet at right angles. The diagram given in the question is the way to approach it and using the simple circle theorem to relate the two angles together as described in the model solution.


OK so yeah I do know the notation of a, b and c etc.

You do say that use the circle theorems which I will.

A few questions; what were you trying to say with that link? Since I have shown a worked example can you please show what you mean as I think it's fair as I've made an attempt.
In particular I wanted to know what you meant by this below for future reference.
Original post by mqb2766
In
https://www.mathopenref.com/lawofsinesproof.html
the line h is the altitude. Its OP in your posted question. So drop a line from a vertex so that its perpendicular to the opposing side. Then you can prove the sin rule by using basic trig on the two right triangles to calculate h and equate.
Original post by Mad Man
OK so yeah I do know the notation of a, b and c etc.

You do say that use the circle theorems which I will.

A few questions; what were you trying to say with that link? Since I have shown a worked example can you please show what you mean as I think it's fair as I've made an attempt.
In particular I wanted to know what you meant by this below for future reference.

You asked about this question in a thread a few days ago
https://www.thestudentroom.co.uk/showthread.php?p=97686111
so you have the solution. In this question, they want you to relate each z/sin(Z) ratio to twice the circumcircle radius. You can use this to prove the sin rule, but its a bit more than usually required as you don't need to know that z/sin(Z) is twice the circiumcircle radius.

For the link
https://www.mathopenref.com/lawofsinesproof.html
the prove the sin rule for a couole of side-angle ratios by dropping a single altitude without any reference to the circumcircle. Then prove its equal to the third side-angle ratio by dropping a different altitude. Its the most direct way to prove the basic sin rule without reference to the circumcircle and the 2R property.

If you dont understand either proof, try and be clear about which parts as Im beginning to lose the thread of what youre asking aobut.
Reply 16
Original post by mqb2766
You asked about this question in a thread a few days ago
https://www.thestudentroom.co.uk/showthread.php?p=97686111
so you have the solution. In this question, they want you to relate each z/sin(Z) ratio to twice the circumcircle radius. You can use this to prove the sin rule, but its a bit more than usually required as you don't need to know that z/sin(Z) is twice the circiumcircle radius.

For the link
https://www.mathopenref.com/lawofsinesproof.html
the prove the sin rule for a couole of side-angle ratios by dropping a single altitude without any reference to the circumcircle. Then prove its equal to the third side-angle ratio by dropping a different altitude. Its the most direct way to prove the basic sin rule without reference to the circumcircle and the 2R property.

If you dont understand either proof, try and be clear about which parts as Im beginning to lose the thread of what youre asking aobut.

Thank you. I know I asked this before. Overall I think I need to as you said brush up on some circle theorems and maybe even angle rules etc.
Original post by Mad Man
Thank you. I know I asked this before. Overall I think I need to as you said brush up on some circle theorems and maybe even angle rules etc.


It honestly shouldnt take more than a day or so. Dropping an altitude is useful for deriving both the sin and cos rules and the circle theorems have relatively simple proofs based on isosceles triangles as you think about triangles with sides where some of them are radii. Note again for this thread, that these proofs wont be evaluated in the proof section, theyre about understanding the trig section as above.
Original post by Mad Man
Thank you. I know I asked this before. Overall I think I need to as you said brush up on some circle theorems and maybe even angle rules etc.


From a quick google, the E1 section in tlmaths has some decent videos (not surprisingly) on deriving the sin and cos rules along the lines mentioned above
https://www.tlmaths.com/home/a-level-maths/full-a-level/e-trigonometry/e1-trigonometry

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